The power series expansions of logarithmic Sobolev, $\mathcal{W}$- functionals and scalar curvature rigidity

Abstract

In this paper, we obtain that the logarithmic Sobolev and $\mathcal{W}$-functionals admit remarkable power series expansions when appropriate test functions are selected. Using these expansions formulas, we prove that for an open subset $V$ in an $n$-dimensional manifold $M$ with $\bar{V}\subset M$ satisfying: (a)The scalar curvature of $V$ satisfies the lower bound:$$\mathrm{Sc}(x)\ge n(n-1)K\text{for all }x\in V,$$ (b) The isoperimetric profile of $V$ is no less than that of space form $M_K^n$:$$I(V,\beta ):=\underset{\begin{array}{c}\Omega \subset V\\ \mathrm{Vol}(\Omega )=\beta \end{array}}{\inf }\mathrm{Area}(\partial \Omega )\ge I(M_K^n,\beta )\text{for some }{\beta }_0>0\text{ and all }0<\beta <{\beta }_0,$$\textbf{then} the sectional curvature of $V$ must satisfy $$\mathrm{Sec}(x)=K\text{for all }x\in V.$$ Additionally, we derive some new scalar curvature rigidity theorems concerninglogarithmic Sobolev inequality and Perelman's $\mathbf{\mu }$-functional.

arXiv commentThis update modifies only the document layout and English phrasing for improved clarity and readability. All mathematical content and theorems remain unchanged from the previous version (Mon, 16 Sep 2024)